Unphysical Magnetic Field Parity: Two Impossibility Directions for Hawking-Induced Magnetism
Author: J. Landers
Abstract
We analyze whether a magnetic field generated by Hawking radiation near a Kerr black hole can ever
reach parity with a plasma / accretion-disk–generated magnetic field. We present two complementary
mathematical results. First, a forward “no-parity” theorem: a bounded spin-dependent factor and a tiny
physical prefactor impose a strict ceiling on the Hawking-induced contribution, preventing equality with
the plasma field over the admissible domain. Second, an inverse non-identifiability theorem in a
poloidal–toroidal (PT) spherical-harmonic decomposition: even from complete measurements of the aggregate
field, the Hawking component cannot be stably recovered; in information-theoretic terms, the measurement
channel has effectively zero capacity for that component. Together, these establish a two-sided
impossibility: no parity in forward models and no recovery in inverse models.
1. Context and Setup
Let $a^* \in [0,1]$ denote the dimensionless Kerr spin, $M$ the mass, $r_+=M\!\left(1+\sqrt{1-(a^*)^2}\right)$
the horizon radius (geometric units), and $\Omega_H=\dfrac{a^*}{2Mr_+}$ the horizon angular velocity.
On physical grounds, the Hawking-induced magnetic field amplitude near the horizon carries a factor
$$
g(a^*) \;\equiv\; a^*\sqrt{1-(a^*)^2}, \qquad 0 \le g(a^*) \le \tfrac12,
$$
multiplying a very small prefactor $C\ll 1$ determined by constants and microscopic efficiencies.
By contrast, a plasma/disk dynamo field increases with accessible rotational energy and does not inherit
the same universal geometric cap.
Intuition. At small $a^*$, the Hawking side appears to grow (quadratically in $a^*$), but the
$\sqrt{\,1-(a^*)^2\,}$ factor enforces a hard maximum at $a^*=1/\sqrt2$. The plasma side is limited by
astrophysical conditions, not by a Kerr-geometry cap.
2. Forward Direction: No-Parity Theorem
Definition (Parity).
On a domain $D\subseteq[0,1]$ in spin (or an equivalent evolution parameter), the Hawking-induced field
$B_{\mathrm{H}}(a^*)$ and the plasma field $B_{\mathrm{pl}}(a^*)$ achieve parity if
$\exists\,a^*_0\in D$ such that $B_{\mathrm{H}}(a^*_0)=B_{\mathrm{pl}}(a^*_0)$.
Theorem 1 (No parity under admissibility cap).
Let $D=[0,a^*_{\max}]$ with $0<a^*_{\max}\le 1$. Suppose
$$
B_{\mathrm{H}}(a^*) \;=\; C\,g(a^*), \qquad g(a^*)=a^*\sqrt{1-(a^*)^2}, \qquad 0\le C\le C_{\max},
$$
and the plasma field obeys a monotone lower bound
$$
B_{\mathrm{pl}}(a^*) \;\ge\; A(a^*) \;\;\text{on } D,
$$
for some increasing function $A$ with $A(a^*_{\max}) \gt C_{\max}\,\max_{a^*\in D} g(a^*)$.
Then $B_{\mathrm{H}}(a^*) < B_{\mathrm{pl}}(a^*)$ for all $a^*\in D$; in particular, parity is impossible on $D$.
Proof.
For all $a^*\in D$, $B_{\mathrm{H}}(a^*)\le C_{\max}\max_{D} g = C_{\max}g_{\max}$ with
$g_{\max}\le \tfrac12$. By hypothesis,
$A(a^*_{\max}) > C_{\max}g_{\max}$ and $A$ is increasing, hence $A(a^*)\ge A(a^*_{\max}) > C_{\max}g_{\max} \ge B_{\mathrm{H}}(a^*)$ for all $a^*$.
Since $B_{\mathrm{pl}} \ge A$, we obtain $B_{\mathrm{H}}(a^*)<B_{\mathrm{pl}}(a^*)$ on $D$. ∎
This is a simple “ceiling vs. ramp” argument: if one curve is globally capped and the other grows past
that cap somewhere in the admissible window, they never meet.
Corollary 1 (Linear lower envelope).
If $B_{\mathrm{pl}}(a^*)\ge \alpha\,a^*$ on $D=[0,a^*_{\max}]$ with $\alpha>0$, then a sufficient condition for no parity is
$$
C_{\max}\,\max_{a^*\in D} g(a^*) \;<\; \alpha\,a^*_{\max}.
$$
Since $\max g=\tfrac12$ (attained at $a^*=1/\sqrt2$), any choice with $C_{\max}/2 < \alpha\,a^*_{\max}$ suffices.
Corollary 2 (ΩH-form).
If $B_{\mathrm{pl}} \propto \Omega_H$ and $B_{\mathrm{H}}\propto (r_+)^{-1}/\sqrt{1-(a^*)^2}$, then equating
$B_{\mathrm{H}}=B_{\mathrm{pl}}$ eliminates $r_+$ and reduces to
$$
a^*\sqrt{1-(a^*)^2} \;=\; \text{constant} \times C,
$$
i.e., the same capped geometric factor appears. With $C\ll 1$, no admissible $a^*$ yields parity.
3. Inverse Direction: Non-identifiability in a PT–Spherical-Harmonic Basis
On a sphere $r=R$ outside the horizon, decompose the divergence-free magnetic field via the poloidal–toroidal
(PT) representation:
$$
\mathbf{B}(R,\theta,\phi)
\;=\;
\nabla\times\big(T(R,\theta,\phi)\,\hat{\mathbf{r}}\big)
\;+\;
\nabla\times\nabla\times\big(P(R,\theta,\phi)\,\hat{\mathbf{r}}\big),
$$
with $T,P$ expanded in spherical harmonics $Y_{\ell m}$. Let the modal coefficients be
$T_{\ell m}(R), P_{\ell m}(R)$, and split each as a sum of plasma and Hawking contributions:
$$
T_{\ell m}=T^{(\mathrm{pl})}_{\ell m}+T^{(\mathrm{H})}_{\ell m},\qquad
P_{\ell m}=P^{(\mathrm{pl})}_{\ell m}+P^{(\mathrm{H})}_{\ell m}.
$$
Theorem 2 (Non-identifiability & zero-capacity bound).
Assume for every $(\ell,m)$ the Hawking modal amplitudes are bounded by
$$
\big|T^{(\mathrm{H})}_{\ell m}\big|,\ \big|P^{(\mathrm{H})}_{\ell m}\big| \;\le\; C_{\ell m},
$$
with $C_{\ell m}$ proportional to $C\,g(a^*)$ (hence extremely small). Suppose the plasma model has uncertainty
$\Delta p_{\ell m}$ (turbulence/modeling/noise) so the measured coefficient $b_{\ell m}$ satisfies
$$
b_{\ell m} \;=\; p^{(\mathrm{model})}_{\ell m} \;+\; \Delta p_{\ell m} \;+\; h^{(\mathrm{H})}_{\ell m},
$$
where $h^{(\mathrm{H})}_{\ell m}$ denotes the true Hawking coefficient. If no prior guarantees
$|\Delta p_{\ell m}| < C_{\ell m}$, then $h^{(\mathrm{H})}_{\ell m}$ is not uniquely recoverable from $b_{\ell m}$.
Moreover, any estimator can certify at most
$$
\big|h^{(\mathrm{H})}_{\ell m}\big| \;\le\; \min\!\big\{\,\big|b_{\ell m}-p^{(\mathrm{model})}_{\ell m}\big|,\ C_{\ell m}\big\}.
$$
Proof.
Fix $(\ell,m)$. For any admissible $h$ with $|h|\le C_{\ell m}$, choose an equally admissible plasma error
$\Delta p = (b-p^{(\mathrm{model})})-h$. Without a prior restricting $|\Delta p|$, both $(h,\Delta p)$ and
$(0,\,\Delta p+h)$ explain the same observation $b$. Hence $h$ is not identifiable; only the bound stated follows by
triangle inequality. ∎
Intuition. The Hawking piece is so tiny that ordinary plasma variability can always “mask” it within the same
mode. In coefficient space, the Hawking vector lies inside the plasma-error uncertainty ball; any putative Hawking
signal can be reinterpreted as plasma jitter.
4. Synthesis and Implications
- Forward impossibility: The geometric cap $g(a^*)\le \tfrac12$ and small prefactor $C$ bound
$B_{\mathrm{H}}$ far below any reasonable lower envelope of $B_{\mathrm{pl}}$ on the admissible domain.
Parity is ruled out (Theorem 1 and corollaries).
- Inverse impossibility: Even with complete spherical-harmonic/PT data, the Hawking component is
non-identifiable under realistic plasma uncertainty; only mode-wise upper bounds are provable
(Theorem 2), implying an effective zero-capacity channel for Hawking magnetism.
Together, these show the Hawking-radiation magnetic field is not merely practically undetectable; it is
structurally excluded in both forward and inverse senses. The magnetic degrees of freedom that
could be conveyed by Hawking quanta are mathematically barred from parity and from stable recovery.
5. Minimal Reference Relations (for completeness)
- Horizon radius: $r_+=M\!\left(1+\sqrt{1-(a^*)^2}\right)$.
- Horizon angular velocity: $\Omega_H=\dfrac{a^*}{2Mr_+}$.
- Spin factor (geometric cap): $g(a^*)=a^*\sqrt{1-(a^*)^2}\le \tfrac12$ (max at $a^*=1/\sqrt2$).